THIRD ELLIPTIC INTEGRAL AND THE ELLlPSOTOMIC PROBLEM. 281 



in the form 



N COS 26 = (u + q 4- 2(j') ^/(u* + 2qu - <f - 2qq') (11), 



N sin 20 = (u 4- 1q 4- q 1 ) x /(- 2 - '2q'u + 2^' + r/ 2 ) (12), 



N 2 = (,,'- r,)3 (</ + , y ) (13), 



so that the condition 



a [udu 



O = \ u (14), 



U == (ir 4- 2qu r/ 2 2qq')( 2 2q'u + 2qq' + q' 2 ) (15), 

 is satisfied by taking 



t=a _ _, H_ 2 , /2 ft, = ._ 8 '/ , A 



A degenerate circular orbit if f/ is obtained by taking 



q + q' = Q, it = 0, ii =0, H 2<//r, 



P = //'-</, ^ ~ ^'/ ( l7 )- 



But the integral in ' (Euvres,' 2, p. 147, putting .'-' + '/ = n, requires /x = 0, /it,, = 0. 

 so that we merely obtain a conic for P = /A,/ 1 . 



40. 



IM _. f - P () (8 - *) + Q () cte 



- 





( a + 2 + a 3 - x 2 )' 



. . i 1 + a + a 2 (1 a) x x~ , l Y / 1 \ 



= 5 Sln 2 l 3 2V V t "** ^/' 



(a + a 2 + a d a;*)' 



L X 2 = a 3 (1 4- a + 2 ) 1(1+ ) ;i (1 ~ ) * ~ ^ ( 2 )' 



P ( v ) = JL a - (1 - a) (5 + 3a + 3a 2 + a") = ar,zn/K' (3), 



Q () = i (i _ 4) (a 4. fl > 4. a s) = jCjS^n/R'cn/K' dn/ K' (4), 



VOL. CCIIT. A. 2 O 



